How would you encode your input vector/matrix from a sequence of moves in game like tasks to train an AI? e.g. Chess AI?
The core of the question seems to really be: "how to approach thinking about this", where "this" is the input of an AI player.

Modern attempts at game playing AI players try to replace a human player "as is". No advantage whatsoever. This implies that we want to feed the same "raw input" to the software player and to the human player. For a video game, the raw input is usually the screen we see, but it could also include sound, vibration feedback from a game controller, force feedback from driving gears, etc. Basically any sensor input available. For a physical board game like Go, the raw input could be a video feed, as done for AlphaGo. Doing so, we fall back to the video game approach. So the input to the AI player is often a tensor, where each element is the intensity of a screen pixel, or some other sensor signal (note there is nothing here about the shape of the tensor, please see below).

The way of thinking here is actually pretty simple, and neat: We want an AI player to replace a human player. So we list up what the human player can sense, and model each input as a tensor (this includes vectors and matrices).

There is perhaps a subtlety here. We face choices not so obvious to make, and there are usually trials and errors (many errors). For example, the Deep-Q Network from DeepMind to play Atari games takes as input differences between two consecutive video frames. Some other approach could solve the same problem with actual sequences of inputs (e.g. 10 consecutive frames). The two approaches are valid---they just have different tradeoffs we must evaluate to find the best configuration.

Another subtelty is the "shape" of the input. Should a single (W, H) input frame be a (W*H) vector, or a (W, H) matrix, or something else? The ramifications can go far. If the frame is an RGB frame with transparency, the input could well be a tensor of shape (W, H, 4), a "cubic volume" with 4 "slices" of shape (W, H)---one for each RGB channel, and a fourth one for transparency. Imagine now if you also have sound, and can add a fifth "slice"---and more if you have stereo or Dolby system channels.

Last point: There is no reference to the architecture of the AI player beyond the input here. Although most exemples refer to recent architectures based on neural networks, this "thinking" also applies whatever can leverage the input. For example, the Big Blue system from IBM, who won Chess against Gary Kasparov, is an AI player using exclusively (if my memory's good) clever tree search. Yet its input is a tensor representing the Chess board.